Regarding the relationships among different insurance claims, especially in non-life insurance, the dependence behaviour in various models has been studied extensively. In this thesis, some discrete-time risk models with dependence structures would be investigated. One traditional discrete-time risk model is the time series risk model, in which the dependence would be on two aspects: time correlated claims and dependent business classes. A general vector (multivariate) autoregressive moving average (VARMA) model would be adopted to analyze the ruin probability of a surplus process. An upper bound for the ruin probability is derived for the general order of multivariate time series models in claims. Simulation studies are carried out for model comparison for finite time ruin probabilities. Another class of risk model is the compound binomial risk model, where the dependence structure would be based on the existence of a so-called by-claim in the claim process. The by-claim could be incurred in the same period as the main insurance claim, or it would be incurred in the next period, depending on a certain probability. A randomized dividend payment scheme with some fixed threshold value in surplus level would also be considered in this thesis. A methodology is discovered to obtain the Gerber-Shiu expected penalty function for the extended model. The final model investigated in this thesis is the periodic time series risk model. The periodic structure of the model gives a practical interpretation of the business cycle, in which there are high season and low season for the business. Some lower order periodic time series models are considered for the claim structures. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

In the actuarial literature, dependence structures in risk models have been extensively studied. The main theme of this thesis is to investigate some discrete-time risk models with claim numbers modeled by integer-valued time series processes. The first model is a common shock risk model with temporal dependence between the claim numbers in each individual class of business. Specifically the Poisson MA(1) process and Poisson AR(1) process are considered for the temporal dependence. To study the ruin probability, the equations associated with the adjustment coefficients are derived. Comparisons are also made to assess the impact of the dependence structures on the ruin probability. Another model involving both the correlated classes of business and the time series approach is then studied. Thinning dependence structure is adopted to model the dependence among classes of business. The Poisson MA(1) and Poisson AR(1) processes are used to describe the claim-number processes. Adjustment coefficients and ruin probabilities are examined. Finally a discrete-time risk model with the claim number following a Poisson ARCH process is proposed. In this model, the mean of the current claim number depends on the previous observations. Within this framework, the equation for finding the adjustment coefficient is derived. Numerical studies are also carried out to examine the effect of the Poisson ARCH dependence structure on several risk measures including ruin probability, Value at Risk, and conditional tail expectation. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy

(Uncorrected OCR) Abstract of the thesis entitled ON INSURANCE RISK MODELS WITH CORRELATED CLASSES OF BUSINESS submitted by Wu Xueyuan for the degree of Doctor of Philosophy at The University of Hong Kong in February 2004 In this thesis, we focus on ruin analysis of risk models wIth correlated classes of insurance business. Specifically, five risk models with different dependence relations between classes are introduced. For these models, various problems related to ruin probability are considered. vVe first study a continuous-time correlated aggregate clmms model with Poisson and Erlang risk processes. In this model, we assume that two classes of business are correlated through a common Erlang component in thelf claim-number processes. We derive an explicit expression for the mfimte-time survival probability of the assumed model when claim SIzes are exponentially distributed. For general claim-size distributions, we obtain some result for the infinite-time ruin probabIlIty, and present a numerical method for evaluating the probability of rum. Based on the continuous-tIme model of Yuen and "Vang (2002) with thin- ning correlatIOn, we propose a new dependence relatIOn with interaction between classes of business in the discrete-time case. Two dIscrete-time risk models with such a relation of dependence are studied. For the first interaction model: we investIgate the statIstical properties of the aggregate claIms for a family of claimnumber distributions. \Ve also compare the model with other existing models with correlated aggregate claIms in terms of the finite-time and infimte-time ruin probabllitles. The second model extends the interaction dependence to the case of the compound binomlal model with delayed claims. For this model, we develop a recursive method to compute the finite-time survival probabilities: and derive an explicit expression for the infinite-time survival probability in a special case. The last two risk models proposed in this thesis are the bivariate compound binomial model and the bivariate compound Poisson model. In the bivariate case: vanous definitions of ruin can be considered. For the bivariate compound binomial model, recursive algorithms for calculating several kinds of finite-time survival probability are presented and numerical examples are given. As for the bivariate compound Poisson model, we study the probabllity that at least one of the two classes of business will get ruined. Since this bivanate ruin probability is very dlfficult to deal with, we use the result of the bivariate compound binomial model to approximate the desired bivanate finite-time survlval probability. \Ve also obtain an upper bound for the infinite-time ruin probability via some association properties of the model. For a simplified version of the model, we examine 'l'l the mfimte-time ruin probability when claIm sizes are exponentially distributed. / abstract / toc / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

Please read the abstract in the section 00front of this document / Dissertation (MCom (Actuarial Science))--University of Pretoria, 2005. / Insurance and Actuarial Science / unrestricted

Credibility theory is widely used in insurance. It is included in the examination of the Society of Actuaries and in the construction and evaluation of actuarial models. In particular, the Buhlmann credibility model has played a fundamental role in both actuarial theory and practice. It provides a mathematical rigorous procedure for deciding how much credibility should be given to the actual experience rating of an individual risk relative to the manual rating common to a particular class of risks. However, for any selected risk, the Buhlmann model assumes that the outcome random variables in both experience periods and future periods are independent and identically distributed. In addition, the Buhlmann method uses sample mean-based estimators to insure the selected risk, which may be a poor estimator of future costs if only a few observations of past events (costs) are available. We present an extension of the Buhlmann model and propose a general method based on a linear combination of both robust and efficient estimators in a dependence framework. The performance of the proposed procedure is demonstrated by Monte Carlo simulations.